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43ISSN 2G72-59B1
Volume ll, No. l, pages 7-l3, 2009
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Editors-in-Chief Jean Jeener (Universite Libre de Bruxelles, Brussels) Boris Kochelaev (KSU, Kazan) Raymond Orbach (University of California, Riverside)
Executive Editor
Yurii Proshin (KSU, Kazan) [email protected]
Editors
Vadim Atsarkin (Institute of Radio Engineering and Electronics, Moscow) Detlef Brinkmann (University of Zurich, Zurich) Yurij Bunkov (CNRS, Grenoble) John Drumheller (Montana State University, Bozeman) Mikhail Eremin (KSU, Kazan) Yoshio Kitaoka (Osaka University,
Osaka)
Boris Malkin (KSU, Kazan) Haruhiko Suzuki (Kanazawa University, Kanazava) Murat Tagirov (KSU, Kazan)
*
In Kazan State University the Electron Paramagnetic Resonance (EPR) was discovered by Zavoisky E.K. in 1944.
3+ 3+
Determination of tetragonal crystalline electric field parameters for Yb and Ce ions from experimental g-factors values and energy levels of Kramers doublets
A. S. Kutuzov*, A. M. Skvortsova
Kazan State University, Kremlevskaya, 18, Kazan 420008, Russian Federation *E-mail: [email protected] (Received October 2, 2009, accepted October 22, 2009)
The tetragonal crystalline electric field parameters for Yb3+ and Ce3+ ions are expressed via ground multiplet exited doublets energies and parameters defining doublets’ wave functions. The crystalline electric field parameters for Yb3+ ion in YbRh2Si2, YbIr2Si2 and KMgF3 crystals extracted from excited state doublets energies and g-factors of ground state doublet are compared with parameters determined in other works.
PACS: 75.10.Dg, 76.30.-v, 75.20.Hr
Keywords: crystalline electric field parameters, g-factors, Yb-based intermetallides, heavy-fermion systems
1. Introduction
Our work was initially stimulated by investigation of heavy-fermion Kondo lattice compounds. Very peculiar magnetic, thermal and transport properties of 4/-electron based heavy-fermion systems are determined by the interplay of the strong repulsion of 4/-electrons on the rare-earth ion sites, their hybridization with wide-band conduction electrons and an influence of the crystalline electric field. The main features of the electron paramagnetic resonance (EPR) signal observed in YbRh2Si2 and YbIr2Si2 [1, 2] (anisotropy of the g-factor and the EPR linewidth) and static magnetic susceptibility [3] of these compounds reflect local properties of the Yb3+ ion in the crystalline electric field (CEF).
In this paper we present the detailed calculation of CEF parameters from energies of ground multiplet exited Kramers doublets and g-factors of ground state Kramers doublet. Our results could be applied to the entire classes of compounds with Yb3+ and Ce3+ tetragonal centers.
2. Diagram of Yb3+ g-factors
A free Yb3+ ion has a 4/13 configuration with one term 2F. The spin-orbit interaction splits the 2F term into two multiplets: 2F7/2 with J = 7/2 and 2F5/2 with J = 5/2, where J is value of the total momentum J = (Jx, Jy, Jz). Multiplets are separated by about 1 eV [4]. As the spin-orbit coupling is much stronger than the CEF in the case of rare earth, we will consider only the ground multiplet 2F7/2 with states | J = 7/2, MJ) = | MJ ), where MJ is the eigenvalue of Jz, z is the tetragonal axis. The Hamiltonian of the Yb3+ ion interaction with the tetragonal CEF could be written via equivalent operators Oqk (J) [4]:
H = aBO + p( b4 o40 + BO4)+r(B0o66 + b64o64) , (1)
where Bq are the CEF parameters, a= 2/63, p= -2/1155, y= 4/27027 [4].
As follows from the group theory, the two-valued irreducible representation D7/2 of rotation group contains two two-dimensional irreducible representations r7 and r6 of the double tetragonal group: Din = 2r'7 + 2r'6 [4].
Therefore the states of Yb3+ in the tetragonal CEF are four Kramers doublets. As the decomposition of D72 includes
twice each of representations r'7 and r6, the matrix of operator (1) could be expressed via two two-dimensional matrices
2C1 C3 ^ (2 A A3 }
1 3 I and I 13 I, (2)
c3 2C2 J 1 A3 2 A2 J’
the former corresponding to bases |5/2), |—3/2) and |—5/2), |3/2), the latter corresponding to bases |7/2), |—1/2) and |—7/2), |1/2). It is convenient to introduce parameters C, A and D:
C = C -C2 = 4B20/21 + 40B40/77-560B60/429, A = A, -A2 = 4B2/7 + 8B40/77 + 80B60/143, (3)
D = -C -C2 = A1 + A2 = 2B0/21 -64B40/77 -160B60/429 ,
where Ci + C2 + Ai + A2 = 0 as traces of Oq are equal to zero. C3 and A3 are
C3 =-8V3B44/77-8^V3B64/1287 , A3 =-8V35B44/385 + 80a/35B4/3003 . (4)
Table 1. Energies, wave functions and g-factors o/ Yb3+ ion in tetragonal crystalline electric field.
E12 = -D ± C/cos%7 E34 = D ± A/cos%6
|1r7 t,^) = +Cl | ±5/2)±c-2 | +3/2) |2r7 t,^) = +c21 ±5/2)±c-1 | +3/2) |3r6 t,^) = ±fl1| +7/2)±a2| ±1/2) 14r6 t,^) = + a2 |+7/2)±a1 | ±1/2)
g (1’2 r7 ) = gJ (5c22 - 3c22,1 ) = gJ (1 ± 4 cos %7 ) gx (1’2 r7) = +4^gJc1c2 = +^V3gJ sin%7 g (3’4 r6 ) = gJ (a22,1 - 7a22 ) = -gJ (3 ± 4cos %6 ) g x (3’4 r6) = -4 gJa21 = -2 gJ (1 + cos%)
Let us define eigenvectors of matrices (2) (c12 , ±c21) and (a12 , ±a21) via angular parameters ^7 and ^6 which correspond to r7 and r6 symmetries: c1 = cos(^7 / 2), c2 = sin(^7 / 2) and a1 = cos(^6 / 2), a2 = sin(^6 / 2). Since matrices (2) are diagonal in the bases of their eigenvectors we can find the relations between our angular parameters and CEF parameters: tan ^7 = C3 /C, tan y6 = A3 /A, it is enough to take —n/2 < ^7, ^6 < n/2. The eigenenergies Ek , wave functions
and g-factors of Kramers doublets are given in table 1. In this table kr7 and kr6 are symmetry symbols, where k = 1..4
is the number of Kramers doublet. The arrow \ or { in wave functions corresponds to the upper or lower sign and denotes up and down effective spin projection. They have been chosen such that <|| J+ ||) 4 0, where J+ = Jx + iJy. Moreover, the phases of the wave function have been chosen as 0||) = | j), where 0 is a time reversing operator [4]. In g-factors left and right indexes correspond to the upper and lower signs; gJ = 8/7 is the Lande g-factor.
The Zeeman energy gJ ^BHJ in the basis ||), | j) of each doublet could be represented by matrix
HZm = g^HzSz + gx^B (HA + HySy ) , (5)
where
g = 2gJ <Vz|t) , gx= gJ <V+|^> , (6)
and H is the magnetic field, S is the effective spin operator with S = 1/2 , /uB is the Bohr magneton, gy and gx are
g-factors when the field is applied parallel and perpendicular to the tetragonal z-axis, respectively (tab. 1).
In the case of cubic symmetry B20 = 0 , B44 = 5B40 and B64 =-21B60 , so that tan%7 = -\/3 , tan%6 = -V35 , c1 =\f3/2 , c2 = -1/2, a1 = V7/12 , a2 = -J5/12 . In accordance with expansion T8 =r7 +r6 [4] the doublets 2r7 and 3r'6 merge into a cubic quartet r8 with energy E(r8) = -16B4°/77 + 1280B60/429 . The doublets 1 r7 and 4r'6 turn into cubic doublets r7 and r6 with energies E(r7) = 144B40/77-320B60/143 and E(r6) = -16B40/11 -1600B60/429
and with isotropic g-factors g(r7) = 3gJ = 3.429 and g(r6) = —7/3gJ = —2.667, respectively. Here r6,78 are irreducible
representations of double cubic group [4].
As g-factors of each doublet depend only on one parameter ^6 or ^7 (tab. 1) we can find the equation relating g|| and gx. Figure 1 represents the diagram of g-factors. The solid and dashed parts of the line g||+ 2gx+ 7gJ = 0 correspond to the doublets 4 r'6 and 3 r'6 , the solid and
dashed parts of the ellipse
(gH - gJ )2/4 + gX/3 = 4 gJ correspond to the doublets
2 r'7 and 1 r'7 . The line and
the ellipse meet in the point (—gJ , —3gJ) marked by a star.
On the diagram (fig. 1) we marked experimental
Figure 1. The diagram of g-factors of Yb3+ ion in tetragonal crystalline electric field values of Yb3+ g-factors in
and experimental g-points taken from literature (tab. 2). several crystals (see also
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
4
3
2
1
0
-1
-2
-3
-4
-5
Table 2. Experimental g-factors of Yb + ion in tetragonal crystalline electric field given in figure 1.
YbRh2Si2 [1] YbIr2Si2 [2] HfSiO4 [5] KY3Fj0 [6] KMgF3 [7] CaF2 [8]
| gll| 0.17(7) 0.85(1) 6.998(6) 5.363(5) 1.070(1) 2.412(3)
| g±| 3.561(6) 3.357(5) 0.4(3) 1.306(2) 4.430(3) 3.802(5)
tab. 2). This allows us to estimate the signs of g-factors and to make assumptions about the ground state Kramers doublet on the basis of measured absolute values of g-factors.
For example, it is evident that the ground state doublet of Yb3+ ion in HfSiO4 is 3 r'6 and both parallel and perpendicular g-factors have a negative sign (if we choose the positive sign in Zeeman energy as in (5)). The ground state doublet of Yb3+ ion in KMgF3 is 4 r'6, the sign of g|| is positive, the sign of g± is negative. In CaF2 crystal the tetragonal center of Yb3+ is in state 1 r7, and the sign of g is positive but the sign of g± can be both positive and negative (two points on fig. 1). In KY3F10 the absolute values of g-factors have such values that do not allow to select the ground state between 3 r6 and 1 r7. In YbRh2Si2 and YbIr2Si2 crystals g± can also be both positive and negative if ground state doublet is 2 . But 4 rj could be considered as candidates for the ground state. A slight difference
between experimental and theoretical values can be explained mainly by taking into account the Kondo interaction, i.e. an exchange coupling between the 4f-electrons of the Yb3+ ion and conduction electrons [3].
3. Calculation of CEF parameters for Yb3+ ion. Comparison with another papers.
Let us calculate the CEF parameters for the given exited state doublets energies A1 < A2 < A3 . It follows from (3) that
B20 = 3A/2 + C/2 + D/2, B40 = A/16 + 5C/16 -D , B60 = 39A/160 - 91C/160 - 13D/40 (7)
and from (4) that
B44 =-7V35A3/16 - 35V3C3/16,
B64 = 117V35A3/160 - 273^C3/160 .
(8)
Taking one of the doublets with energy Ek (tab. 1) as the ground, defining the differences of doublets energies as Emk = Em - Ek and solving this system of linear equations we can express C, A and D through Emk . Substituting relations A3 = A tan ^6 and C3 = C tan into (8) and then C, A and D into (7) and (8) we find:
13 1
B2 = 8b + 4 b6 cos% + 4b7 cos^7 ,
B40 =-1 b +—b6 cos®, +—b7cos®7,
4 4 32 6 32 7
D0 13 , 39 , 91,
B6 =-------b +-b6 cos®,-------b7 cos®7,
6 160 320 6 320 7
D4 7a/35, . 3 5^3, .
B4 =-----32" b6 sin^6-32" b7 sin^7 ,
D4 117>/35, . 273V3, .
B6 =---------b6 sin®6-----------b7 sin®7.
6 320 6 320 7
(9)
where b, b6 and b7 are determined in table 3. To use (9) we have to choose the ground state doublet and the exited state
Table 3. b, b6 and b7 in (9).
doublets sequence to express energy differences Emk in table 3 through experimental values A1 < A2 < A3. Angular parameters ^6 and ^7 can take the values -n/2 < ^6, ^7 < n/2 independently, the energy scheme does not depend on them. To determine the values of ^6 and ^7 we have to use additional experimental results. Thus the experimental values of ground state Kramers doublet g-factors can help us to define the ground state using figure 1 and one of the angular parameters: ^6 in the case of r'6 ground state doublet symmetry or ^7 in the case of r7 ground state doublet symmetry. But the sign of this angular parameter remain undefined. For doublets with r6 symmetry it happens because g|| and g± depend only on cos ^6 (tab. 1), and for doublets with r7 symmetry the reason is that in usual EPR experiments we are able to define only the absolute values of g-factors, therefore we have to consider two points on g-diagram (fig. 1) with opposite sings of gL ~ sin ^7 (tab. 1). Notice that only B44 and B64 in (9) depend on the signs of ^6 and ^7.
Ground state b b6 b7
1 rt 7 E31 ^ E21 + E41 E31 ^ E41 -E21
2 r E32 ^ E12 + E42 E32 ^ E42 E12
3 r'6 E43 ^ E13 ^ E23 E43 E13 — E23
4 rt 1 6 E34 ^ E14 ^ E24 E34 4 2 E2 14
Table 4. g-factors values from figure 2 and corresponding values of parameters y6 or (see tab. 1).
Compound g-point g|| g± Ref. ^6 or p7
YbRh2Si2 1 - 0.17 - 3.561 [1]*
2 - 0.18 - 3.846 [10]
3 - 0.169 - 3.794 [9] <p7 = - 1.2798
4 - 0.169 - 3.916 [9] ^6 = ± 0.7769
5 - 0.307 - 3.847 [9] cp6 = ± 0.8191
6 - 0.20 - 3.897 [10] ^6 = ± 0.787
YbIr2Si2 7 - 0.85 - 3.357 [2]*
8 - 0.918 - 3.626 [10]
9 - 0.887 - 3.558 [10] = ± 0.9811
KMgF3 10 1.070 - 4.430 [7]
11 0.992 - 4.496 [7] ^6 = ± 0.2576
* g-factors absolute values measured at 5 K
We have compared our results with [9], [10] and [7]. In these papers the CEF parameters for Yb3+ ion in YbRh2Si2 [9,10], YbIr2Si2 [10] and KMgF3 [7] crystals were calculated with the use of least squares method, i.e. authors tried to find CEF parameters which give best coincidence between numerically calculated and experimental values of ground state doublet g-factors and energy levels. Figure 2 and table 4 represent experimentally measured and theoretically calculated g-factors from [9], [10] and [7]. CEF parameters obtained in these papers are given in table 5.
In [9] YbRh2Si2 compound has been investigated (fig. 2a). Using the least squares method the absolute values of g-factors (tab. 2) and energies of three excited levels (17, 25 and 43 meV [11]) have been taken into account. All obtained sets of CEF parameters (tab. 5) satisfy exactly the experimental energy scheme of 2F7/2 multiplet and give negative signs of g and gL (points 3, 4 and 5 on fig. 2a), i.e. correspond to the lowest point from two points for given crystal on figure 1.
In the case of r'7 symmetry of ground state doublet CEF parameters calculated by authors of [9] (tab. 5) correspond to point 3 on figure 2a, but it is not the closest point to the experimental one. CEF parameters from [9] could be obtained from our expressions (9) for Ai, ^6, and doublets sequence given in table 5. Considering the case of r'6 ground state doublet symmetry the authors of [9] note that the mean values of experimental g-factors <|g|> = (|g||| + 2|gi|)/3 = 2.43 are closer to the absolute value of cubic r6 doublet g-factor (g = 2.67) than to the absolute value of cubic r7 doublet g-factor (g = 3.43) . However, we have to notice that taking into consideration the signs of g-factors, the point g = gL = (g> = -2.43 lies almost on the ellipse corresponding to doublet 2r'7 on figure 1. This doublet 2 r'7 is not originated from the cubic doublet r7 but appears to be a result of the cubic quartet r8 splitting (see above). Moreover, the g-curve of doublet 2 r7 is closer to the experimental g-point than g-line of doublet 4 r6. The CEF parameters calculated by authors of [9] for the case of r6 ground state doublet symmetry correspond to the optimal point 5 on the figure 2a (the values of parameters are not given in [9]).
g| | g| | g| |
Figure 2. Experimentally measured (tab. 2) and theoretically calculated g-factors of Yb3+ ion in (a) YbRh2Si2, (b) YbIr2Si2 and (c) KMgF3 from [9], [10] and [7]. Numerical values of numbered g-points are given in table 4.
Table 5. Comparison of Yb3+ ion CEF parameters in YbRh2Si2, YbIr2Si2 and KMgF3 crystals from [9], [10] and [7] with parameters calculated from (9). CEF parameters Bqk, parameter of the spin-orbit interaction £ and
exited state doublets energies Ai are given in meV.
Compound YbRhfSi* YbIrfSi* KMgF3
Reference [9] Eq. (9) [9] Eq. (9) [10] Eq. (9) [10] Eq. (9) [7] Eq. (9)
B20 11.7 11.73 25.0 24.92 21.70 21.74 2.75 2.78 105.38 105.56
B40 -7.4 -7.4 1.9 1.83 -0.02 -0.02 5.18 5.17 4.84 4.58
B44 77.6 77.62 46.0 45.64 51.88 51.79 42.10 42.13 157.95 152.6
B0 -4.0 -3.98 1.7 1.61 4.92 4.93 8.64 8.63 -0.124 -0.125
b4 -18.5 -18.52 -60.5 -60.04 -56.33 -56.22 -33.01 -33.05 16.98 17.52
£ 359.8 360.03
96 -1.2818 -0.7769 -0.787 -0.9811 0.2576
97 -1.2798 -0.4525 0.9557 0.4634 0.9135
Doublets sequence 2, 4, 1, 3 4, 2, 1, 3 4, 1, 2, 3 4, 1, 3, 2 4, 2, 1, 3
g-point 3 4 6 9 11
A1 A2, A3 17, 25, 43 18, 25, 36 13.14, 87.28, 125.59
Besides, the authors of [9] have calculated CEF parameters for Tj, ground state doublet symmetry case taking into account all states of 2F term and therefore considering both crystal field and spin-orbit interaction with spin-orbit interaction constant as an additional fitting parameter. The g-factors values calculated in [9] correspond to the point 4 on the figure 2a which still lies on the line we have plotted considering only ground 2F7/2 multiplet. CEF parameters (tab. 5) are also well reproduced by expressions (9).
In [10] YbRh2Si2 and YbIr2Si2 crystals have been considered. In the frame of the least squares method authors took into account only states of ground multiplet 2F7/2 , the experimental values of energies (17, 25 and 43 meV for YbRh2Si2 [11] and 18, 25 and 36 meV for YbIr2Si2 [12]) and g-factors (see point 1 on fig. 2a and point 7 on fig. 2b) increased at 8 % (see point 2 on fig. 2a and point 8 on fig. 2b). The authors argue that this increase of the absolute values of g-factors is caused by the interaction with conduction electrons. r'6 symmetry doublet was considered as ground state. The theoretical g-points found in [10] are the optimal points 6 and 9 (fig. 2a,b). The corresponding CEF parameters coincide with those calculated from expressions (9) (tab. 5).
In paper [7] CEF parameters of Yb3+ ion in KMgF3 crystal have been found (tab. 5). Using the least squares method the experimental values of g-factors (tab. 2) and experimental energy of whole 2F term levels have been taken into account. Obtained CEF parameters satisfy the experimental energy scheme of 2F term very well, but are reproduced by our expressions (9) only approximately (tab. 5), because we have found these expressions taking into account only ground multiplet 2F7/2. Experimental g-points 10 and theoretical g-points 11 corresponding to CEF parameters from [7] are given on fig. 2c. It is remarkable that point 11 lies on the line g|| + 2g± + 8 = 0 which we have plotted considering only the ground multiplet 2F7/2. This can be explained as follows. Expressing wave functions of ground state doublet r'6 in term of ionic states | J, Mj ) of 2F term as |t, ^) = ±p1 | 7/2, + 7/2) ± p2 17/2, ± 1/2) + p3 | 5/2, ± 1/2) where pf + p22 + p32 = 1 we find that
„ 64 2 8^3 62 2 32 2 W3 18 2
g|| =-8 + — p2----— p2 p3 + — Pз, gX=-T p2 +~ p* p3 + ~T p*. (10)
In this case g|| and g± are related by the equation g + 2gL + 8 = 14p32, but as the admixture of excited 2F5/2 multiplet is small (p3 = 0.00551 [7]) we obtain previous relation g + 2g± + 8 = 0.
Note that consideration of experimental energy levels of whole 2F term for YbRh2Si2 and YbIr2Si2 crystals could eliminate the uncertainty in CEF parameters determination (9).
4. CEF parameters for Ce3+ ion.
The ground multiplet of free Ce3+ ion is 2F5/2 and the excited multiplet 2F7/2 has energy greater for 273 meV [4]. Let us consider ground multiplet 2F5/2 with states | J = 5/2, MJ ) = | MJ ), where MJ is the eigenvalue of Jz. The Hamiltonian of the Ce3+ ion interaction with the tetragonal CEF could be written via equivalent operators Oq (J) [4]:
Table 6. The energies, states and g-factors of Ce3+ ion in tetragonal crystalline electric field.
E12 = D ± A/cos® E3 = -2D
|1r7 t,^) = a1 | ±5/2) + a2 | +3/2) |2r'7 t,^) = a2 | ±5/2)-a1 | +3/2) |3r6 t,^) = | ±1/2)
g|i C1’2 r7) = gj (5al,2 - 3a22,i) = gj (1 ± 4cos®) g (3 r'6) = gj
g 1C1’2 r7) = ±2^5gja^2 = ±45gj sin® g± (3 r'6) = 3gj
H = aB000 + P(B000 + B44044),
(11)
where Bq are the CEF parameters, a= -2/35, p = 2/315 [4].
The two-valued irreducible representation D5'2 of rotation group contains two two-dimensional irreducible representations r7 and r of the double tetragonal group: D5 2 = 2r7 +r'6 [4]. Therefore the states of Ce3+ in the tetragonal CEF are three Kramers doublets. The decomposition of D5 2 includes once r and twice r7 representations. The doublet with |±1/2) states and energy (±1/2|H |±1/2) corresponds to r representation. To find energies and states corresponding to r'7 representation we have to diagonalize two-dimentional matrices
2A1 A3
A3 2 A2
(12)
of operator (11) on bases |5/2), |—3/2) and |—5/2), |3/2). It is convenient to use parameters A, D and A3:
A = A1 - A2 = -12B0/35 +16B0/21, D = A1 + A2 = -8B0/35 -8B0/21, A3 = 8a/5B44/105 .
(13)
The eigenvectors of matrix (12) (a12 , ±a21) could also be written via angular parameter: a1 = cos(p / 2), a2 = sin(p / 2). Diagonalizing the matrix (12) we find that tan p = A3 /A, —n/2 < p < n/2. The energies Ek (k = 1..3), states ||), ||) and g-factors of Kramers doublets are given in table 6. In this table the arrow t (I) and the left (right) index correspond to the upper (lower) sign; gj = 6/7 is the Lande g-factor.
For the r'7 doublets the gy and g1 are related by expression (g - gj)2/16 + g 1/5 = g2. The left and right parts of ellipse constructed in (gy , g1) axis would correspond to 2 r'7 and 1 r7 doublets, respectively.
Let us define the CEF parameters for given energies of exited state doublets A1 < A2. It follows from (13) that
B0 =-5 A/ 4 - 5D/ 2,
B0 = 3 A/ 4 - 9D/ 8,
B44 = 105A3/8yf5.
Choosing one of the doublets with energy Ek (tab. 6) as a ground state, solving system of linear equations Em we can express A and D through Emk . Substituting relation A3 = A tan p and then A and D into (14) we can find:
(14)
Em Ek
B2 =— b +— b cos®, 2 12 8
B40 =— b — b cos®, 4 16 8
B 4 =- J05_ B4 16V5
b sin®,
(15)
Table 7. b and b in (15).
Ground state b b
1 rt 7 21 — E3 2 E21
2 r 2 3E 2 — 1E 2 — E12
3 r6 3 E1 1 3 2 3 E1 1 3 2 E2
where b and b are determined in table 7. To use (15) we have to choose the ground state doublet and the exited state doublets sequence to express energy differences Emk in table 7 through experimental values A1 < A2. The value of angular parameter p in (15) lies within interval —n/2 < p < n/2, the energy scheme does not depend on it. To define the
value of p it is necessary to use other experimental data. In the case of
r'7 ground state doublet symmetry the experimental values of g-factors could help to define p. However, as the sign of g-factor cannot be defined from usual EPR experiment and g1 ~ sin p (tab. 6), so the sign of p and therefore the sign of B44 in (15) stay
undefined.
In the case of cubic symmetry B0 = 0, B44 = 5B0, so tan® = V5/2 , a1 =-J5/6 , a2 =-J 1/6 . The doublets 1 r'7
and 3 r'6 merge into a cubic quartet r8 with energy E(r8) = 16B0/21 in accordance with expansion T8 = r7 +T'6 [4].
The doublet 2 r7 turns into a cubic doublet r7 with energy E(r7) = -32B0 /21 and with isotropic g-factor g(r7) = —5/3gj = —1.429. Here r78 are irreducible representations of the double cubic group.
5. Summary
For Yb3+ and Ce3+ ions all possible sets of tetragonal crystalline electric field parameters that satisfy the given experimental energy scheme of ground multiplet are defined.
For Yb3+ ion the CEF parameters (9) beside the energies of Kramers doublets depend also on angular parameters p6 and p7 , defining wave functions of r'6 and r7 symmetry doublets correspondingly. Their values are undefined and lie within the interval —n/2 < p6 , p7 < n/2 independently. To define these parameters exactly it is necessary to use another experimental set of data. For example, experimental absolute values of ground state doublet g-factors allow to define the absolute value for one of angular parameters.
The earlier published CEF parameters for Yb3+ ion in YbRh2Si2, YbIr2Si2 and KMgF3 crystals calculated with the use of least squares method could be obtained from our formulas (see tab. 5).
For Ce3+ ion the CEF parameters (15) beside the energies of Kramers doublets depend also on angular parameter —n/2 < p < n/2, defining wave functions of r'7 symmetry doublets. In case of r'7 ground state doublet symmetry the experimental absolute values of g-factors could help to define the absolute value of this parameter.
Acknowledgments
We wish to acknowledge professor B.I. Kochelaev for bringing our interest to this problem and continuous interest to the results.
This work was supported by the Volkswagen Foundation (I/82203) and partially by the RF President Program ‘Leading scientific schools’ 2808.2002.2; AMS was supported partially by grant Y4-P-07-15 from the CRDF BRHE Program; ASK thanks the program "Development of scientific potential of a higher school" (Number of grant: 2.1.1/2985) for the partial support.
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